Pentagon

A Maths Starter Of The Day

On a full page in the back of your exercise book draw a perfectly regular pentagon.

This image is a work of a
U.S. military or Department of Defense employee, taken or made during the course
of an employee's official duties.
As a work of the U.S. federal government, the image is in the public domain.

I like the construction task.However, the origami method does not produce a regular pentagon, but one which is very close.The angle formed by the first two folds is in fact 109.47 degrees and not 108. This can be established through the fact that the ratio of sides of A4 paper (or A3, A2 etc) is always 1 to root 2, and similar triangles. Hence the tangent of half of the internal angle is root 2.I became suspicious through understanding that it is impossible to construct a regular pentagon without a protractor.

Mark Richer, Brockworth Enterprise Scool

Following my comment of 9 Dec where I refuted the Origami method of producing a regular pentagon from a sheet of A4 paper, and stated that I did not believe one could be constructed without a protractor, I was delighted to view on your web-page a graphical demonstration of a non-protractor method which showed how a pentagon could be constructed.My instinct tells me that this is not possible. I have checked out the method shown, using traditional Pythagorean and Trigonometrical methods, using a standard calculator giving results to 8 or 9 d.p., I found lengths and angles to be within 40 parts per million of the theoretical values, which is well within the bounds of rounding errors in the complex series of calculator processes. This leaves me in two minds.1. Do I eat ‘humble pie’ and accept that my assumption was wrong? No angle-relationships or other mathematical methods known to me can prove that this construction method is valid.2. Just like the Origami method which I know is flawed, could it be that some clever person has come up with another contrived construction method which ‘produces the goods’ in practice, but is flawed in theory?

Mr Mark Richer, Brockworth Enterprise School

Are comments from teachers on the 'Starter of the Day' website vetted?Is there a mechanism by which mathematical statements which are clearly wrong can be deleted so that both pupils and teachers are not misguided?I of course refer to the suggestion that a regular pentagon can be constructed.I am concerned that for as long as this page shows an incorrect method (or even two), we are not informing learners, but arming them with invalid mathematical ideas.

Transum,

Mark, thanks so much for your comments. We are just about to leave for the Christmas holidays but when we return will remove the word 'perfect' from the origami film. We will happily take down any of your previous comments if you request us to but they do make an interesting read!

How did you use this starter? Can you suggest
how teachers could present or develop this resource? Do you have any comments? It is always useful to receive
feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.Click here to enter your comments.

If you don't have the time to provide feedback we'd really appreciate it if you
could give this page a score! We are constantly improving and adding to these
starters so it would be really helpful to know which ones are most useful. Simply
click on a button below:

Excellent, I would like to see more like this
Good, achieved the results I required
Satisfactory
Didn't really capture the interest of the students
Not for me! I wouldn't use this type of activity.

Answers

The internal angles of a regular pentagon are 108o
and the external angles are 72o. The following diagram shows a way to
constrict a pentagon without a protractor:

Did you know that you can fold a regular pentagon from a sheet of A4 size paper?

If you can't see the images above it may be because you do not have a recent version of the Flash Player installed or that your device (such as the iPad) cannot show Flash diagrams.

Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items chosen and recommended by Transum Mathematics to get you started.

Have you read Craig's book yet?

Craig Barton must surely be the voice of Mathematics teachers in the UK. His wonderful podcasts interviewing the industry experts have culminated in this wonderful book. As Craig says: "I genuinely believe I have never taught mathematics better, and my students have never learned more. I just wish I had known all of this twelve years ago..." more...

"How I wish I’d taught maths' is an extraordinary and important book. Part guide to research, part memoir, part survival handbook, it’s a wonderfully accessible guide to the latest research on teaching mathematics, presented in a disarmingly honest and readable way. I know of no other book that presents as much usable research evidence on the dos and don’ts of mathematics teaching in such a clear and practical way. No matter how long you have been doing it, if you teach mathematics—from primary school to university—this book is for you." Dylan Wiliam, Emeritus Professor of Educational Assessment, UCL.

Casio Classwiz Calculator

There is currently a lot of talk about this new calculator being the best in its price range for use in the Maths classroom. The new ClassWiz features a high-resolution display making it easier to view numerical formulas and symbols but it isn't a graphical calculator as such (it has the capacity to draw graphs on your smart phone or tablet, via a scannable QR code and an app).

As well as basic spreadsheet mode and an equation solving feature you also get the ability to solve quadratic, cubic or quartic polynomial inequalities and the answer is given just as it should be written down, using the correct inequality symbols!

This calculator has a high-performance processor and twice the memory of previous models ensuring speedy operation and superior computational power.more...

Teacher, do your students have
access to computers?Do they have iPads or Laptops in Lessons?

Whether your students each have a TabletPC, a Surface or a Mac,
this activity lends itself to eLearning (Engaged Learning).